xSkill: notes, lectures and practice for CBSE and CSE

1.1 The wave toolkit Waves carry energy and momentum through a medium without the medium travelling as a whole. Almost every question in this chapter connects three quantities: the wave speed, the wavelength and the frequency. The formulas below keep that connection in each setting, from a stretched string to an organ pipe. 1.2 How the sheet is organised The sheet follows the order in which the chapter is built. The first block gives the progressive wave, because every later result starts by writing a disturbance as a sine of a phase. The second block collects the three speed formulas: one for a stretched string, one for a fluid, and one for a gas after the Laplace correction. The third block handles standing waves, first on strings and then in pipes, and the final block adds beats and the Doppler effect. When a numerical question arrives, decide which block it belongs to, write the matching relation, and only then substitute numbers. 1.3 Progressive wave

y (x, t) = A sin (k x - ω t + ϕ) k = \frac{2 π}{λ}, ω = 2 π f

Equation of a sinusoidal progressive wave

v = \frac{λ}{T} = λ f = \frac{ω}{k}

Speed, wavelength and frequency

y = f (x - v t) travels along + x, y = f (x + v t) travels along - x

Positive and negative direction of travel

Δ ϕ = \frac{2 π}{λ} Δ x Δ x = \frac{λ}{2 π} Δ ϕ

Phase and path difference

1.4 Speed of waves

v = \frac{T}{μ} μ = \frac{m}{L}

Transverse wave on a stretched string

v = \frac{B}{ρ} (fluid) v = \frac{Y}{ρ} (rod)

Longitudinal wave in a fluid and in a solid rod

v_{Newton} = \frac{P}{ρ} v_{Laplace} = \frac{γ P}{ρ} = \frac{γ R T}{M}

Speed of sound: Newton and Laplace

v_{t} = v_{0} + 0.61 t (v \propto T)

Temperature dependence of the speed of sound in air

1.5 Standing waves

y = 2 A sin (k x) cos (ω t) nodes at k x = nπ

Standing wave from two opposite travelling waves

λ_{n} = \frac{2 L}{n} f_{n} = \frac{n v}{2 L}, n = 1, 2, 3, \dots

Standing waves on a string fixed at both ends

open pipe: f_{n} = \frac{n v}{2 L} closed pipe: f_{n} = \frac{( 2 n - 1 ) v}{4 L}

Organ pipes

e = 0.6 r L_{effective} = L + e (open end)

End correction for a pipe of radius r

1.6 Units and signs Every quantity in the sheet has a fixed unit. Wavelength is in metres, frequency in hertz, angular frequency in radians per second, wave number in radians per metre, tension in newtons and linear density in kilograms per metre. The phase constant does not decide the direction of travel; the signs of the x and t terms do. The Doppler signs must be chosen from the physical approach or separation before any substitution. When a question gives centimetres, convert to metres before using a formula with metres in it, then convert the answer back if the options are in centimetres. 1.7 Beats and Doppler

f_{beat} = ∣ f_{1} - f_{2} ∣ f_{avg} = \frac{f _{1} + f _{2}}{2}

Beat frequency

f^{'} = f \frac{v \pm v _{o}}{v \mp v _{s}}

Doppler effect for sound (source and observer along the line)

Exam

The upper signs in the Doppler formula go with motion towards the other body. A moving observer changes the number of wavefronts met per second, while a moving source changes the wavelength in front of it, so the two effects are not symmetric.

Identify the wave type before choosing a speed formula.
Convert wavelength to metres before dividing by the period.
Use the string formula v = sqrt(T/mu) with the tension in newtons.
Remember that a closed pipe has only odd harmonics.
Use the beat frequency as the difference of frequencies, never their sum.
Write the Doppler signs from the physical approach or separation, then check the result.